 Schreier refinement theorem

Any two finite subnormal series have refinements with the same successive factors up to ordering and group isomorphism. For descending chains $A_i$ and $B_j$, insert $A_{i+1}(A_i\cap B_j)$ between $A_i$ and $A_{i+1}$ and insert $B_{j+1}(B_j\cap A_i)$ in the other chain. The Zassenhaus butterfly lemma identifies the factor indexed by $(i,j)$ in the first refinement with the factor indexed by $(j,i)$ in the second. If both chains are composition series, their simple factors admit no nontrivial refinement, giving the Jordan–Hölder theorem.

The Schreier refinement theorem is a result in group theory that deals with the relationship between subgroups and normal series of a group. It provides criteria for refining a normal series of groups, allowing for more structured decompositions of groups into simpler components. The theorem is primarily used in the study of group extensions and solvable groups.